C₅×#12172 #11934892

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12172

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#2190	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

344 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,17)(9,12)(10,25)(11,16)(13,19)(14,21)(15,23)(18,31)(20,30)(22,26)(24,27)(28,37)(29,36)(32,34)(33,35);(7,12)(9,21)(10,23)(13,26)(14,15)(16,27)(17,31)(18,34)(20,35)(22,36)(24,37)(25,28)(29,30)(32,33);(6,8)(7,13,16,17)(9,22)(10,20)(11,19)(12,26,27,31)(14,29,28,32)(15,30,25,33)(18,24)(21,36)(23,35)(34,37)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5160);